Write the sum $LaTeX: \displaystyle -42-43-44 \ldots -65-66$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-43-(-42)=-1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -42+(n-1)(-1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -42+(n-1)(-1)=-66 \implies n = 25$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{25} \left(- n - 41\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 25(-42-66) }{2}=-1350$ .